There are numerous branches of mathematics, from the familiar arithmetic, geometry and algebra at an elementary level to more advanced branches such as number theory, topology and complex analysis. Each branch has its own distinct set of axioms, or fundamental assumptions, from which theorems are derived by logical processes.
While each has its own flavour, character and methods, there are also strong overlaps and interdependencies between sub-fields. Several attempts have been made to construct a grand unified theory that embraces the entire field of maths.
Set theory, introduced by Georg Cantor, provides a foundation for mathematics. In principle, all the theorems of maths can be derived by starting from the axioms of set theory, known as the Zermelo-Fraenkel axioms. Another unifying framework is category theory, which considers structure-preserving mappings between general objects. However, this has not gained the traction enjoyed by set theory.
There are some spectacular examples of unification. Most notable is the development of analytic geometry, or co-ordinate geometry, which stands in contrast to synthetic geometry, the usual Euclidean approach.
The Cartesian co-ordinate system of René Descartes introduced the now-familiar system of x and y axes, enabling geometric objects such as lines and circles to be defined by algebraic equations. By analysing these equations, purely geometric results can be derived by algebraic processes.
Another example of an unexpected connection is the prime number theorem, giving the density of prime numbers. This was proved using methods from analysis, a branch of mathematics not obviously related to number theory. This led on to the emergence of a new field, analytic number theory.
Letter from jail
André Weil was one of the giants of 20th-century mathematics. While in prison in 1940, as a result of refusing military service, Weil formulated some conjectures about deep connections between number theory and geometry. In a letter to his sister, the noted philosopher Simone Weil, he explained his ideas in clear and simple language.
Canadian mathematician Robert Langlands wrote to Weil, suggesting much more sweeping interconnections. He modestly wrote that, if Weil was not interested in the contents, "I am sure that you have a waste basket handy".
The ideas set out in the letter led to the “Langlands program”, a set of deep conjectures that attempt to forge links between algebraic and analytical objects. Great progress has been made in proving these conjectures, but most of them remain open. Langlands proposed that many problems in number theory could be solved by the methods of harmonic analysis, a completely different field of maths. This implied strong connections between fields that were believed to be unrelated, and some of these have since been found.
Another focus of number theory is Diophantine equations, polynomial equations with integer coefficients, where we are interested in integer or whole-number solutions. Perhaps the most famous Diophantine equation is Fermat's equation. Fermat's Last Theorem was proved in the 1990s by Andrew Wiles, who employed methods of algebraic geometry. Wiles' work is a spectacular example of a merger of geometry and number theory, and falls within the ambit of the Langlands program.
Patterns emerging in disparate areas can indicate a mysterious underlying structure of all mathematics. The Langlands program is like a Rosetta Stone of mathematics, as yet undeciphered. Many mathematicians believe it holds the key to understanding the essential nature of mathematics.
Peter Lynch is emeritus professor at UCD School of Mathematics & Statistics – he blogs at thatsmaths.com